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w = x u

The vorticity w of a fluid flow is simply the curl of the velocity field u.
The vorticity can be thought of as measuring the local rotation of the fluid: locally we can regard the flow as being the sum of a translational, a pure straining and a rotational part, and the angular velocity in the last component is w/2.

We can write the Navier-Stokes equation in the form

u/∂t + w x u = (p/ρ + u2/2 + G) + v∇2u

where p is the pressure, ρ is the density, G is the gravitational potential and v is the kinematic viscosity. Taking curl of both sides gives

w/∂t + (u.)w - (w.)u + w(.u) - u(.w) = v∇2w

We assume that the fluid is incompressible, so .u = 0. Since the divergence of curl always vanishes .w = 0 too. Hence we obtain the vorticity equation

w/∂t + (u.)w = (w.)u + v∇2w

Each of the last three terms represent the three ways that vorticity changes: convection, strengthening from stretching, and diffusion.
If we assume that the fluid is inviscid (i.e. v = 0) then we can write the vorticity equation as

Dw/Dt = (w.)u

where D/Dt is the substantial derivative operator.
With the vorticity equation expressed this way it is immediately obvious that if the fluid flow is initially irrotational (i.e. w ≡ 0) then the flow will remain irrotational for all time in the absence of viscous effects. This is a very useful result since irrotational flow can be dealt with rather effectively. It allows the introduction of a velocity potential, so finding u reduces to solving the quite well-understood Laplace equation, and with that we get some nice uniqueness results in the bargain.

Vorticity therefore is what makes fluid dynamics difficult, and understanding it is one of the major challanges facing a fluid dynamicist.

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